Lemma 103.10.3. Let $f : \mathcal{X} \to \mathcal{Y}$ be a flat morphism of algebraic stacks. Then $Q_\mathcal {X} \circ f^* = f^* \circ Q_\mathcal {Y}$ where $Q_\mathcal {X}$ and $Q_\mathcal {Y}$ are as in Lemma 103.10.1.

**Proof.**
Observe that $f^*$ preserves both $\mathit{QCoh}$ and $\textit{LQCoh}^{fbc}$, see Sheaves on Stacks, Lemma 96.11.2 and Proposition 103.8.1. If $\mathcal{F}$ is in $\textit{LQCoh}^{fbc}(\mathcal{O}_\mathcal {Y})$ then $Q_\mathcal {Y}(\mathcal{F}) \to \mathcal{F}$ has parasitic kernel and cokernel by Lemma 103.10.2. As $f$ is flat we get that $f^*Q_\mathcal {Y}(\mathcal{F}) \to f^*\mathcal{F}$ has parasitic kernel and cokernel by Lemma 103.9.2. Thus the induced map $f^*Q_\mathcal {Y}(\mathcal{F}) \to Q_\mathcal {X}(f^*\mathcal{F})$ has parasitic kernel and cokernel and hence is an isomorphism for example by Lemma 103.9.4.
$\square$

## Post a comment

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like `$\pi$`

). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.

## Comments (0)